The prefix af or (rarely) the more modern spelling av, is common for Swedish noble family names. The word litterally means "of" in this case. Examples: "af Ugglas". The German "von" is also common, e.g. "Carl von Linnaeus"

The only rule for these prefixes in Sweden is that only noble families can have them. An ordinary citizen can't pop an "af" or a "von" into his name because it would look good.

To make things complicated: the use of these prefixes is a relatively late invention. The old Swedish noble families often had short names, usually corresponding to a prominent design on the family coat of arms, e.g. "Grip" (Griffin) or "Bielke" (bar). Sometimes, though, noblemen added the name of their estate after the family name, using "af" or "till" as a preposition.